This work aims at developing a reliable and efficient strategy to simulate electromagnetic stirring (EMS) applications in steelmaking industry, especially for ...
MAGNETOHYDRODYNAMICS Vol. 53 (2017), No. 3, pp. 547–7
Numerical coupling strategy for the simulation of electromagnetic stirring L. Marioni1,2 , E. Hachem2 , F. Bay2 1
Transvalor S.A. 694 Avenue Doct. Maurice Donat, Mougins, France MINES ParisTech, Center for Materials Forming (CEMEF), UMR CNRS 7635, BP 207, 06904 Sophia Antipolis, France
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This work aims at developing a reliable and efficient strategy to simulate electromagnetic stirring (EMS) applications in steelmaking industry, especially for in-mould stirring (M-EMS) in continuous casting and for ingot casting processes. Several advanced numerical tools have been used to enhance the simulation capabilities and a coupling numerical scheme has been proposed to couple the different physics involved in the simulation. This coupling strategy has been developed in a commercial software focused on casting simulations. Finally, the model has been tested and compared to experimental and numerical results present in the literature.
This is the author’s copy of the accepted manuscript. Please refer to: L. Marioni, F. Bay, E. Hachem. Numerical coupling strategy for the simulation of electromagnetic stirring. Magnetohydrodynamics, 53(3), 547-557, 2017 for the edited/published version.
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Introduction Electromagnetic stirring is widely used to increase the efficiency in steelmaking processes such as continuous and ingot casting. By controlling the molten metal flow it is possible to regulate temperature variation, avoid segregation and achieve better microstructure in the final product. Since laboratory-scale tests are not fully representative of the process and industrial measurements are both expensive and difficult to carry out, numerical simulation is a fundamental tool to study and optimize these electromagnetic applications in steel industry. The main drawback for numerical simulations in this field is that the application to be modelled is a set of complex multiphysics coupled phenomena, thus a direct simulation approach leads to huge computational costs, incompatible with industrial needs. For this reason, this work aims at exploring simulation techniques and setting up a robust and efficient coupling strategy to simulate stirring applications. R The proposed strategy has been implemented in commercial software (THERCAST ). The fluid flow has been modelled by the variational multiscale formulation (VMS) which allows modelling the effects of small scale turbulence without explicitly tracking it, and anisotropic remeshing algorithms have been used to optimize the mesh. The electromagnetic problem has been modelled by the (A, φ) potential formulation and solved in the frequency domain. Finally a weak coupled 2-meshes2-solvers approach has been used to model the coupled problem. 1. Mathematical model
1.1. Electromagnetic problem. The (A, φ) potential model has been used to model the electromagnetic field (EMF). Thus, the evolution in time of the EMF is describes by the following
system: ∂A −1 ∇ × µ ∇ × A = σ −∇φ − ∂t + u × ∇ × A ∂A ∇ · σ −∇φ − + u × ∇ × A = 0. ∂t
(1)
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where µ is the magnetic permeability, A is the magnetic vector potential, φ is the scalar electric potential, u is the medium velocity, and σ is the electrical conductivity. The convective term can be neglected from the first equation in system (1) because the Reynolds magnetic number is Rem 1. The system has been discretized in space by Nedelec edge finite elements [1] and then solved in the frequency domain. For further information about the resolution technique of the resulting complex system via real-equivalent formulation, we refer to [2]. 1.2. Mechanical problem. The turbulent fluid flow has been modelled by the incompressible Navier-Stokes equation: ( ρ (∂t u + u · ∇u) − ∇ · (2ηε(u) − pI) = f g + f L (2) ∇ · u = 0.
where ρ, η, f g and f L are the density, the viscosity, the gravity force, the Lorentz force respectively. I is the identity tensor, while ε is the strain rate tensor defined as: ε(u) :=
1 ∇u + ∇T u . 2
(3)
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System (2) has been solved by finite elements in a variational multiscale approach, as proposed in [3]. According to the multiscale paradigm, the variables have been split into a coarse and a fine scale. The fine scale is not solved directly, but its effects are integrated in the coarse scale resolution. The obtained formulation is both computationally light (because it does not track explicitly the small scale phenomena), and accurate (because it takes into account the effect of the small turbulence scale into the flow), thus it is suitable for highly turbulent flow simulations. For the detailed development of the weak formulation, we refer to [4, 5].
1.3. Coupling scheme. The coupling strategy used in this work is the so called “Two meshes two solvers approach” (2M2S) [6]. The basic principle is that the two problems (i.e. the electromagnetic and the mechanical ones) should be solved by two different software applications on two different domains with different meshes. In this work, the electromagnetic problem has been solved R by using MATELEC in a domain including the molten metal, the mould and the air surrounding R the mould. The mechanical problem has been solved by using THERCAST in a domain coincident with the fluid domain. The Lorentz force is computed in post-process as: ∂A f L = j × B = σ −∇φ − +u×∇×A ×∇×A (4) ∂t
Different techniques to model the convective term have been tested in this work; while its influence on the magnetic field has been neglected (section 1.1), it has been mechanically considered in three different ways: 548
(A) It has been neglected, since Rem 1. The Lorentz force corresponding to the current timestep has been added as a source term in the Navier-Stokes equation. According to [7], the average value of the force has been considered when the frequency f > 2.5I2 , where I2 is the second invariant of the shear rate tensor. (B) The correct value of j has been computed at each time step by solving ∂A ∇ · σ −∇φ − + u × ∇ × A = 0. ∂t
(5)
This value has been used to compute the Lorentz force, but not for re-computing the electromagnetic field.
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(C) It has been considered in explicit, neglecting the divergence constraint (thus considering ∇φ = 0). Hence, we obtain j = −σ (∂t A − u × ∇ × A). The techniques described in B and C use the value of u at the previous time-step, thus the time-step for the mechanical resolution has been chosen according to [8]: 1 h ; , (6) ∆t = min |u| σ|B|2 where h is the mesh size. R R The communication between MATELEC and THERCAST is file-based, thus all the necessary fields (e.g. the Lorentz force, the current density, the magnetic field, depending on the coupling scheme used) are written in files and then read by the following solver. This choice is forced by the fact that we do not base our coupling scheme on the simple passage of one field (e.g. the average Lorentz force), but we need several fields, eventually computed at different moments of the electromagnetic period. Thus, MPI coupling (which is faster) would have been possible only by either coupling the two solvers at each time step or overcharging the RAM, which would have required significantly higher computational resources. In order to guarantee the efficiency of the method for parallel computation, the file based strategy has been applied core-wise: each core writes the results in binary language for its own partition. In this way, it is possible to perform writing and reading processes in parallel, avoiding bottleneck for massively parallel computations. The efficiency in term of computational time has been analysed in section 2.
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1.4. Anisotropic mesh adaptation. The mesh of the mechanical simulation has been adapted anisotropically and dynamically during the simulation. The remeshing algorithm used in this work is based on the error estimation analysis proposed in [9] and aims at producing the optimal mesh to discretize the input fields by minimizing the related interpolation error. For this work, we propose the following set of fields to remesh according to: u |u| d |f | v= ; ; ; (7) |u| maxΩ (|u|) maxΩ (d) maxΩ (|f |) where Ω is the fluid domain, d is the boundary distance and f is the volumetric force. The remeshing algorithm will create the optimal mash which, given a target number of elements, minimises the interpolation error of the fields listed in formula (7) along the elements’ edges. In this way, the elements’ shape is such that it optimally interpolate these fields with the linear shape functions used in the finite elements approach. From the practical point of view,the elements will be stretched perpendicularly to the gradient of these fields. 549
1.5. EM source term definition: Ω condition. The electro-magnetic influence on the fluid mechanical problem is modelled through an explicit source term in the Navier-Stokes equations, as shown in equation (1). This term is evolving in time periodically, but its variation speed may be much higher than the variation rate of the mechanical problem. If the period of this term (proportional to the period of induction T ) is much lower of the turbulence characteristic time, the average value of the Lorentz force may be used. In this case, only one field has to be interpolated and stored, so the coupling interface will be light. On the contrary, if the period of the Lorentz forces is of the same scale as the turbulence period, the variation of the force within the EM period must be considered. This means that the Lorentz force corresponding to each EM time step must be stored and interpolated, leading to an increase in the requested computational effort. The highest period for which no interaction between the transient part of the Lorentz force and the turbulence appears has been calculated by [7] based on the Kolmogorov theory and it is computed as: 1 , 2.5I2
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Tmax =
(8)
where I2 is the second invariant of the shear rate tensor. But this criterion is local. The transient part of the Lorentz force may interact with the turbulence, but not having a global impact or occurring in a region where the force density is so low not to change the inertia of the flow. For this reason, we propose a global criterion, namely the Ω condition to quantify the global interaction between the EM and the CFD problem. Thus, given the condition Ωm n , we want to verify the criterion in equation (8) on n% of the active volume, where the active volume is the region where the Lorentz force is higher than m% of the maximum Lorentz force in any point during the whole EM period. The parameters n and m must be defined for each application. 2. Results
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2.1. Bench Test The numerical framework described in section 1 has been tested with respect to established results proposed in [10]; the bench case consists in a laboratory scale stirring application of Galinstan melt in a Plexiglas mould. The problem configuration and the main parameters are reported in figure 1 and table 1 respectively. Thanks to the axial symmetry of the problem, only a 15◦ -section has been simulated. The electromagnetic simulation has been performed in a domain which includes the inductor, the melt and a air layer 1m tick from the mold external surface. The input current has been set to IRM S = 200 A and the frequency f has been varied from 50 Hz to 1300 Hz. The mesh (figure 2(a)) has been refined isotropically in the melt in order to have at least 12 elements in the skin depth computed as: r
δ=
ρ . πf µ
(9)
The mechanical simulation has been performed in the fluid domain. No slip boundary conditions have been imposed on the solid wall boundaries and perfectly slip boundary conditions have been imposed at the symmetry planes and the free surface. The average number of elements during the dynamic mesh adaptation is 275000. A detail of the mesh used at the steady state of the mechanical simulation is shown in figure 2(b). 550
2.2. Numerical results. Coupling scheme.
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In figure 3, the axial velocity at the centre of the fluid cylinder is plotted. The velocity has been computed at the steady state, i.e. after 200 s and the induction frequency has been set to f = 150 Hz. The results have been compared to both the experimental and numerical results proposed in [10]. In table 2 a summary of the simulation accuracy is provided by reporting the error related to the main result variables: the maximum velocity of the fastest eddy (Ef), the maximum velocity of the slowest eddy (Es) and the position of the separation point between the eddies normalized to the maximum eddies size (Ep). The reference solution is the experimental one provided in [10], which means that a ±10% range of validity has to be accepted for Ef and Es. In figure 4, the maximum recirculation velocity over the induction frequency is plotted. For lower frequencies (where experimental data are available) both VMS and k − ε are in the experimental range; however, for higher frequencies, the two solutions produce different results. In both cases the maximum velocity decreases after a peak, but the VMS solution decreases faster. Also a reference solution (2 × 106 elements) confirms this tendency. Since no experimental data are available in this region, no validation can be affirmed; but the difference in results seems to be a direct consequence of the turbulence model: in VMS a better approximation of small scale turbulence is used, thus higher energy dissipation is produced at high levels of stirring force density.
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The three different coupling techniques described in section 1.3 have been tested; in figure 5, the error of the maximum axial velocity obtained by the different coupling schemes is shown. The error has been computed with respect to the coupling strategy B, which is the most accurate one from the theoretical point of view. Strategy A results to be more accurate than C. In both cases, the error has two peaks: one at low frequency and one at the frequency corresponding to the maximum velocity produced. This is due to the fact that the mechanical effect of the convective term (i.e. u × ∇ × A) depends on two different factors: by increasing the frequency the velocity tends to increase (which increases the effect of the convective term), but the skin depth decreases, dumping the electromagnetic penetration (which decreases the effect of the convective term). It is also remarkable that the approach A leads to an overestimation of the velocity because the braking effect is neglected, while approach C leads to a underestimation of the velocity since the braking term is considered without accounting for the self-balance due to the creation of an electric field ∇φ. In table 3, the time analysis for each coupling scheme is proposed. Let’s consider strategy A as the reference, which takes about 1 day of computation. The resolution of Navier-Stokes equation is the most expensive part of the computation, in particular due to the long physical time to be simulated before reaching the steady state. Approach B, which is the most accurate, is about 50% more expensive than A, since the value of the electrical current has to be computed at each time step by solving the second equation in System (1). Approach C is the less accurate and it is slightly more expensive than A, since the convective term of the Lorenz force has to be computed at each time step. Ω Condition.
The good couple of parameters (m, n) has to be found in order to guarantee the effectiveness of the Ω-condition. Too high values (for instance m = n = 100) would imply that the criterion shown in equation (8) has to be satisfied over the whole domain: in practice, this case never takes place, 551
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so the average Lorentz force would be always used. On the opposite, too small values would lead to the use of the transient value of the force when even only one finite element satisfies criterion (8). In order to find a good couple, different stirring levels has been simulated by increasing the current density in the inductor, until reaching the lever for which the double eddy recirculation pattern is no more developed and a fully turbulent flows occurs. By increasing the stirring, the turbulence energy increases and the turbulence time scale decreases. In this way, larger parts of the domain will exhibit interaction between the turbulence and the transient part of the EM force. In figure 6, the (m, n) diagram is shown for three levels of stirring. For low stirring (figure 6(a)) the axial velocity obtained by the average Lorentz force and the one obtained with the transient part differ of 3%. This value is small, meaning that very little interaction between the EM field and the turbulence occurs. For middle levels of stirring (figure 6(b)), this difference rises to 10%, while for the highest stirring level (figure 6(c)), the difference is 17%. The red region in the diagram represent the set of couples (m, n) for which the Ωm n -condition is satisfied, thus the transient part of the EM force would be used. From figure 6 we can deduce two main trends: • When there is no need of using the transient part of the force, only few couples of (m, n) would lead to take it into account, with the consequent overuse of computational resources. • When there is need of using the transient part of the force, a larger set of couples (coloured area) is acceptable, which makes the choice of (m, n) less delicate. For this specific application, we propose the couple m = 80, n = 5, hence Ω80 5 . In this way, the average value of the force is used until it leads to a 3 − 5% of error for the axial velocity, while the transient part is used above this limit.
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3. Conclusions The coupling strategy between electromagnetic and fluid mechanical simulations has been investigated. Several numerical tools (e.g. VMS, anisotropic remeshing, edge finite elements) have been adopted to achieve a good modelling of both the turbulent flow and the magnetic field produced by the stirrer. The results have been compared to experimental and numerical benchmarks present in the literature; finally, different coupling strategies have been tested in order to determinate the most effective one in terms of accuracy and computational effort. Neglecting the mechanical effect (as well as the electromagnetic influence) of the magnetic field convection is the most efficient choice leading to a small velocity overestimation (Err < 2%) and remarkable computational time decrease (−35% with respect to the mechanical modelling of the convection term).
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5. E. Hachem and B. Rivaux and T. Kloczko and H. Digonnet and T. Coupez. Stabilized finite element method for incompressible flows with high Reynolds number. Journal of Computational Physic, vol. 229 (2010), no. 23, pp. 8643–8665. 6. M. Barna, M. Javurek, J. Reiter, J. Watzinger, B. Kaufmann, and M. Kirschen. Numerical Simulations of the Continuous Casting of Steel with Electromagnetic Braking and Stirring. International Journal of Multiphysics, vol. 2 (2010), pp. 231–238. 7. F. Felten and Y. Fautrelle and Y. Du Terrail and Y. Metais. Numerical modelling of electromagnetically-driven turbulent flows using LES methods. Applied Mathematical modelling, vol. 28 (2004), pp. 15–27. 8. L. Marioni and F. Bay and E. Hachem. Numerical Stability Analysis and Flow Simulation of Lid-Driven Cavity Subjected to High Magnetic Field. Physics of fluids, vol. 28 (2016), 057102.
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9. T. Coupez and E. Hachem. Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing. Computer Methods in Applied Mechanics and Engineering, vol. 267 (2013), pp. 65–85.
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10. D. Musaeva and V. Ilyn and V. Geza and E. Baake. Experimental investigation of lowfrequency pulsed Lorentz force influence on the motion of Galinstan melt. JSt. Petersburg Polytechnical University Journal: Physics and Mathematics, vol. 2 (2016), no. 3, pp. 193–200.
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Figure 1: Geometry of the benchmark case
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Table 1: Bench test main parameters. Parameter Unit Value Galinstan Melt Density Kg/m3 6440 Dynamic viscosity Pa · s 0.0024 Electric conductivity S/m 3.46 · 106 Radius mm 31 Height mm 70 Copper inductor Electric conductivity S/m 1.78 · 108 Number of turns − 6 Turn diameter mm 8 Turns distance mm 4 Inductor radius mm 61
Table 2: Errors related to figure 3 Type of error ANSYS k − ε THERCAST V M S Ef 16.1% 1.4% Es 5.2% 1.0% Ep 7.2% 0.4%
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(a) Mesh used in the electro-magnetic simulation.
(b) Mesh used in the CFD simulation.
Figure 2: Meshes used for the EM and CFD computations 3
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Figure 4: Maximum axial velocity at different frequencies.
Table 3: Computational effort comparison between different coupling strategies. A B C Navier-Stokes resolution 87.3% 56.6% 87% Maxwell resolution 3.6% 2.4% 3.6% Remeshing 8.8% 5.7% 8.7% Coupling interface 0.3% 35.3% 0.7% Total time Ref. +54% +0.3%
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Figure 5: Error of the maximum axial velocity simulated with different coupling strategies.
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(a) Couples of parameters which lead to the use of the (b) Couples of parameters which lead to the use of the transient Lorentz force for low stirring. transient Lorentz force for medium stirring.
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(c) Couples of parameters which lead to the use of the transient Lorentz force for high stirring.
Figure 6: Set of parameters couples for which the transient Lorentz force is used at different stirring intensities.
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